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PropertiesofQuadrilaterals FaSMEd

Subject: MathematicsAgeofstudents: 14-15years

UsedTechnology: Card

Functionalities: Sendinganddisplaying

Time: 1or2lessons

FaSMEdpartner: AfricanInstituteforMathematicalSciences SchoolsEnrichmentCentreShortAbstract: Thislessonisfirstaboutusingpropertiesof

quadrilateralstoidentifyparticularshapesandsecondaboutidentifyingthesmallestnumberofpropertiesneededtodefineashape.Afterashortintroduction,duringwhichtheteacherasksstudentstogivethepropertiesofasquare,thestudentsworkinsmallgroups.Theyaregivenstripsofsmallcards;oneachstrippropertiesofagivenquadrilateralaregiven.Studentssketchthequadrilateralandlaterdeterminethesmallestnumberofpropertiesneededtodefinethequadrilateral.Aftertheactivitytheteacherholdsaclassdiscussionaimingtogoovertheanswersandelicitsomethingaboutthestudents’understanding.

FaSMEdRaisingAchievementthroughFormativeAssessmentinScienceandMathematicsEducation


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1. ContentThislessonisaboutpropertiesofquadrilateralsandidentifyingthesmallestnumberofthesepropertiesneededtodefineashape.

2. Activity2.1 Aims

Thislessonintendedtoembedandreinforceknowledgeaboutquadrilaterals.Inparticular,students

• nameandclassifyquadrilateralsaccordingtotheirproperties• identifytheminimalinformationrequiredtodefineaquadrilateral• sketchquadrilateralswithgivenconditions.

2.2 Structure/Methodology

Timeneeded:60to90minutes

Materialsneeded:

• Miniwhiteboardsforuseintheintroductionandduringthelesson• Stripsofsmallcardsforthestudentstouse(onesetpergroup)• Prestikorblu-tak• Scissors(onepairpergroup)• A3paper(onesheetpergroup)• Instructions(onepergroup)• Bigcardsforuseinthewrap-upwholeclassdiscussion

2.3 Lessonplan

Thisisanabbreviatedlessonplan,adaptedfromtheonefoundathttp://map.mathshell.org/lessons.php?unit=7325&collection=8.

Introduction

Holdaclassdiscussioninwhichyourevisewhatismeantbyaquadrilateralandaskthestudentstogiveyouthepropertiesofasquare.Theymaycomeupwitharangeofpropertiessuchas:

Foursidedshape Twopairsofparallelsides

Closedshape Fourrightangles

Allsidesareequal Twoequaldiagonals

Oppositesidesareparallel Diagonalsbisecteachother


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Discusswhichpropertiesapplyonlytosquaresandwhichapplytooneormoreotherquadrilaterals.(e.g.rectanglesandsquareshavetwoequaldiagonals).

Askthestudentstosuggestonepairofpropertiesthatdefineasquare(e.g.fourrightanglesandfourequalsides).

Remindstudentsoftheconventionsweusetoindicateparallelandequallines,rightanglesandsoon.

Pairwork(Part1)

Askthestudentstoworkinpairs.Toeachpair,giveasheetofA3paper,asetofstripsofcards(seeFigure1),scissorsandaPart1instructionsheet(seeFigure2).

Figure 1: Small strips of cards for students


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ShowthestudentshowtofoldtheirA3sheetofpaperinto6equalparts.

Foreachstripsofcardsstudentsshouldworkoutwhatquadrilateralitdescribes.OntheA3sheetofpaper,theysketchthequadrilateralandnameit.Makesurethattheyleaveplentyofspacebeloweachsketchtostickcardsonforpart2.Studentscanalsobeencouragetoindicatethepropertiesofeachquadrilateraltheydraw.

Asyouwalkaround,listentothestudents’discussionsandaskthemquestionsiftheyappeartobestuck.Avoidtellingthemwhattodo(intermsofthemaths)butaskthemquestionstoleadthemtowardsgettingtotheanswerbythemselves.

Iftimeallows(Part2)

HandoutPart2oftheinstructions(Figure3)forstudents.Theyshouldchooseonestripofcardsand

• cutthelittlecardsout• choosethesmallestnumberof

cardsneededtodefinethequadrilateral

• stickthecardstheychoseunderthesketch.

Theyshoulddothisforallthestrips.

Figure 3: Instructions for students (Part 2)

Figure 2: Instructions for students (Part 1)


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Discussion

Foreachstripofcards,discussthequadrilateralthatwasdefinedaswellasitsproperties.

Ifstudentshavecompletedpart2,askforavolunteertocometotheboard,sketchoneofthequadrilateralsandstickthebigversionsofthelittlecardstheychosetodefinethequadrilateralontheboard.Repeatuntilallthesetsofcardshavebeenused.

Discussthestudents’suggestionsthroughout.Emphasisethatmorethanonesetofcardscanbechosentouniquelydefineaquadrilateral.

2.4 Technology

• Miniwhiteboardsforuseintheintroductionandduringthelesson.

• Setsofsmallcardsforusebythesmallgroupsofstudents

• Setsofbigcardsforuseinthewholeclassdiscussion

Allthesetechnologiesprovideopportunitiesforformativeassessmentinthattheyhelptomakethestudents’thinkingvisible.

2.5 AspectsofFormativeAssessment

Theteachersassessthestudents’levelsofunderstandinginreal-timeandactaccordinglybyaskingquestions,givinghintsandsometimestellingthemtheanswer.Theyclarifylearningintentions,engineereffectivediscussions,providefeedbackandactivatestudentsasresourcesforoneanotherandforthemselves.Thistakesplaceatallphasesofthelesson.

Inthemainactivity,studentsworkinpairstosketchquadrilaterals.Theintentionisthatstudentswilltaketurnstoexplaintheirthinkingtotheothersintheirgroup.Eachstudentshouldengagewiththeirpeer’sexplanationandtakeresponsibilityfortheunderstandingofothersintheirgroup.

Ifthestudentsfollowtheseinstructions,itislikelythattheywillbeassessingtheirpeers’levelsofunderstandingandactingaccordingly,sometimesexplainingtheirownreasoning,sometimestellingthepeerswhattodoandsometimesaskingquestions(i.e.peerformativeassessment).Inthiscasetheywouldbeactivatingtheirpeersasinstructionalresources.

3. FurtherInformation

3.1 Background

ThislessonisbasedontheonedesignedbytheMathematicsAssessmentProject,whichcanbefoundathttp://map.mathshell.org/lessons.php?unit=7325&collection=8.

Thelessondesignincludestheuseofapre-lessonassessmenttask,whichteacherscanusetogatherinformationaboutthecurrentlevelsofstudents’understandingandtheirdifferentproblemsolvingapproaches.Theuseofthistaskishighlyrecommendedbutinpracticemanyteachersdonothavetimetodoso.Apost-lessontaskisalsoprovided.


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3.2 Commonissues

Theissuesarisinginthepre-lessontaskarelikelytoariseinthemaincardactivityandthetableofcommonissuesbelowprovidessomeusefulquestionsandpromptsteacherscouldusetomovethestudents’thinkingon.

Common issues: Suggested questions and prompts:

Understands different types of quadrilaterals as being distinct shapes rather some quadrilaterals being subsets of others For example: The student states that ‘no’ rectangles are squares

• What properties does a rectangle/square have?

• Does a rectangle/square have all the properties of a square/rectangle?

• Is it possible that one type of quadrilateral could be a special kind of a different quadrilateral? How could you tell from the properties if this was the case?

Assumes that the opposite sides of a rhombus are not parallel For example: The student states that ‘no’ rhombuses are parallelograms

Or: The student states that ‘some’ kites are rhombuses

Or: Fails to circle ‘rhombus’ as having at least one pair of parallel sides

• What do you know about the angles in a rhombus?

Assumes that a kite contains parallel sides For example: The student circles ‘kite’ as having at least one pair of parallel sides

• Does a kite have congruent sides? • Which sides in a kite are congruent?

Assumes diagonals that bisect must do so at 90°

For example: The student circles just the square

• What does it mean for diagonals to bisect each other?

Assumes that the diagonals in an isosceles trapezoid bisect each other For example: The student provides an explanation that the diagonals of isosceles trapezoids bisect each other whereas non-isosceles trapezoids contain non-bisecting diagonals

• In what way is an isosceles trapezoid different to a non-isosceles trapezoid?

• Draw in the diagonals of an isosceles trapezoid. What properties would the two triangles that are formed have if the diagonals were bisecting?

Provides little or no explanation For example: The student gives no reason for their choice of word and/or fails to explain their answers

• Which properties of (rectangles) do (trapezoids) not satisfy?

• Can you convince me that a (rhombus) satisfies all the properties of a (parallelogram)?

• What additional properties does a (square) have?


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3.3 Answers

Shape A is a square:

The minimal set of properties contains three cards, for example A2, A3 & A4 define the square.

Shape B is a rectangle:

The minimal set of properties contains three cards, for example B1, B3 & B5 define the rectangle.

Shape C is a parallelogram:

The minimal set of properties contains four cards, for example C2, C3, C4 & C5 define the parallelogram.

Shape D is a rhombus:

The minimal set of properties contains three cards, for example D2, D3 & D5 define the rhombus.

5cm

6cm

4cm

12cm

7cm

125°

125°

55°

55°

7cm110°

70°

110°

70°


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Shape E is a kite:

The minimal set of properties contains four cards, for example E1, E2, E4 & E5 define the kite.

No angles are given for Shape E so when students are sketching the kite they will not be able to label any angles on their sketch. However, it is possible to construct the kite from the information given.

Shape F is an isosceles trapezoid:

The length of the longest side of the trapezoid is not given in the properties of Shape F so students will not be able to label the length of this side on their sketch. However, it is possible to construct the trapezoid from the information given.

All five cards are needed to define the trapezoid.

Note: Some students may sketch shape F as shown below:

This is not possible to draw.

4cm

5cm

2cm

2cm

6cm

10cm

60 60

120 120

6cm

10cm

60

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